Gelfand–Tsetlin patterns and polytopes

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#Gelfand–Tsetlin patterns

Gelfand–Tsetlin (sometimes spelled Cetlin or Zetlin) patterns were introduced by I. Gelfand and M. Tsetlin in [GT50], motivated by the close relationship with representation theory of $\GL_n.$ A short introduction to GT-patterns can be found in [Sta01, p. 103].

For a brief background, see the Gelfand–Tsetlin pattern collection in FindStat [RSo].

I. Fischer and F. Schreier-Aigner prove product formulas for $(-1)$-enumerations of arrowed Gelfand–Tsetlin patterns [FS23]. Their proofs use a recent Littlewood-type identity and determinant evaluations via LU decompositions, Sister Celine’s algorithm, and creative telescoping.

A Gelfand–Tsetlin pattern or GT-pattern is a triangular (or sometimes parallelogram) arrangement of nonnegative integers, with certain constraints. The triangular patterns are indexed as follows. \[\begin{matrix} x_{n1} & & x_{n2} & & \cdots & & \cdots & & x_{nn} \\ & \ddots & & \ddots & & & \iddots & & & \\ % & & x_{13} & & x_{23} & & x_{33}\\ % & & & x_{12} & & x_{22} \\ & & x_{21} & & x_{22}\\ & & & x_{11} \end{matrix}\] The less common parallelogram patterns are indexed as follows. \[\begin{matrix} x_{n1} & & x_{n2} & & \cdots & & \cdots & & x_{nm} \\ & \ddots & & \ddots & & & & & & \ddots \\ % & & x_{13} & & x_{23} & & x_{33} & & \\ % & & & x_{12} & & x_{22} & & & \\ & & x_{11} & & x_{12} & & \cdots & & \cdots & & x_{1m} \\ & & & x_{01} & & x_{02} & & \cdots & & \cdots & & x_{0m} \end{matrix}\] The entries must satisfy the conditions \[x_{i+1,j} \geq x_{ij} \text{ and } x_{ij} \geq x_{i+1,j+1}\] for all values of $i,$ $j$ where the indexing is defined. The inequalities simply state that horizontal rows are weakly decreasing, down-right diagonals are weakly decreasing, and down-left diagonals are weakly increasing.

The weight $w(G) = (w_1,\dotsc,w_n)$ of a GT-pattern $G$ is defined as the vector where $w_i$ is the difference between the sums of rows $i$ and $i-1.$ By convention, $x_{0j}=0$ if these entries are not specified.

#Bijection with semistandard Young tableaux

The conditions ensure every row in the pattern defines an integer partition. Furthermore, any two adjacent rows define a skew shape. These properties put triangular GT-patterns in bijection with semistandard Young tableaux and parallelogram GT-patterns with skew Young tableaux.

The skew shape defined by row $j$ and $j+1$ in a GT-pattern $G$ describes which boxes in a tableau $T$ contain $j.$ In particular, if $T$ has shape $\lambda/\mu$ then the first row in $G$ is $\mu$ and the last row is $\lambda.$

Example

The following triangular GT-pattern \[\begin{matrix} 5 & & 4 & & 2 & & 1 & & 1 & & 0\\ & 5 & & 3 & & 2 & & 1 & & 0\\ & & \color{blue} 3 & & \color{blue} 3 & & \color{blue} 2 & & \color{blue} 1 \\ & & & 3 & & 3 & & 1 & \\ & & & & 3 & & 2 & \\ & & & & & 3 & \\ \end{matrix}\] is in bijection with the semistandard Young tableau below.

$1 $ $ 1 $ $ 1 $ $ 5 $ $ 5$ $ 2 $ $ 2 $ $ 3 $ $ 6$ $ 3 $ $ 4$ $ 4$ $ 6$

Notice that the fourth row in the GT-pattern is $3321$ and the shape of all entries at most four in the tableau is also $3321.$

Example

The skew Young tableau

$ 1 $ $ 3$ $ 1 $ $ 2$ $ 4$ $ 2$

is in bijection with the GT-pattern \[\begin{matrix} 4 & & 3 & & 2 & & 1\\ & 4 & & 3 & & 1 & & 1\\ & & 3 & & 3 & & 1 & & 1\\ & & & 3 & & 2 & & 1 & & 0\\ & & & & 2 & & 1 & & 1 & & 0\\ \end{matrix}\] Observe that in any GT-pattern, $x_{i+1,j} - x_{i,j}$ counts the number of boxes containing $i$ in row $j$ of the corresponding tableau. Thus, the weight of a GT-pattern is the same as the weight of the corresponding tableau.

There is a crystal graph structure on semistandard Young tableaux. The corresponding crystal graph on GT-patterns is described explicitly in [HK26].

#Tokuyama’s formula

Let $SGT(\lambda)\subset \gtp(\lambda)$ be the set of strict Gelfand–Tsetlin patterns. This is the subset of patterns where each row is strictly decreasing. Consider tuples $(a,b,c)$ in the pattern arranged as $\begin{smallmatrix} a && b\\ & c \end{smallmatrix}.$ Let $S(G)$ be the number of entries $c$ such that $a\gt c \gt b,$ and let $L(G)$ be the number of entries $c$ such that $a=c.$

Let $\rho = (n-1,n-2,\dotsc,1,0).$ Then (see [Tok88]) \[\schurS_{\lambda}(x_1,\dotsc,x_n) \cdot \prod_{1 \leq i \lt j \leq n} (x_i + t x_j) = \sum_{G \in SGT(\lambda + \rho)} (1+t)^{S(G)} t^{L(G)}\xvec^{w(G)}.\]

Note that this interpolates between the Weyl character formula and the SSYT definition of Schur polynomials.

See also [Sta86] for some special cases and connections with plane partitions and alternating sign matrices. Y. Alexandr, P. Commins, A. Embry, S. Frank, Y. Li, and A. Vetter study type $B$ analogues via ice models [ACEF+18]. They construct models in bijection with Gelfand–Tsetlin-type patterns and shifted tableaux for the Sundaram rules, obtaining partition functions that deform the Weyl character formula for $SO(2n+1,\setC).$

#Gelfand–Tsetlin polytopes

By specifying the top row in a triangular GT-pattern as $\lambda$ and imposing the inequalities above, we obtain a convex polytope. This is a Gelfand–Tsetlin polytope, $\gtp_{\lambda} \subset \setR^{n(n+1)/2}.$ By construction, the integer lattice points in $\gtp_{\lambda}$ are in bijection with $\SSYT(\lambda).$ The analogous statement holds for parallelogram GT-patterns where the bottom row $\mu$ is also a fixed integer partition. These polytopes are denoted $\gtp_{\lambda\mu} \subset \setR^{nm}.$ Below, we use $d$ to denote the dimension ($n(n+1)/2$ or $mn$) of the ambient space.

#Unimodular triangulation and marked posets

The polytopes $\gtp_{\lambda\mu}$ admit unimodular triangulations. This means that the polytopes can be decomposed into simplices with integer vertices and normalized volume $1.$ For a short proof, see [Ale19].

As a consequence, Gelfand–Tsetlin polytopes have integer vertices, and have the integer decomposition property (IDP). This means that if $p$ is a lattice point in $k \gtp_{\lambda\mu}$ for some integer $k\geq 1,$ then $p$ can be expressed as $p_1 + \dotsb + p_k$ where each $p_i$ is a lattice point in $\gtp_{\lambda\mu}.$ This can be proved directly by bijecting to semistandard tableaux, see [Ale16, Subsec. 2.3]. Under this bijection, elementwise addition of GT-patterns corresponds to concatenating tableaux: a tableau of shape $k\lambda/k\mu$ decomposes into $k$ tableaux of shape $\lambda/\mu.$

A. Tsuchiya generalizes the notion of IDP to $k$-tuples of polytopes (the $k=2$ case was introduced in [HH17]). The following theorem follows directly from the bijection with SSYT.

Theorem (Alexandersson, 2018).

Let $\gtp_{\lambda^1},\dotsc,\gtp_{\lambda^k}$ be GT-polytopes, embedded in the same ambient space by padding the partitions with zeros if needed. Then \[\setZ^d \cap \left( \gtp_{\lambda^1} + \dotsb + \gtp_{\lambda^k} \right) = \left( \setZ^d \cap \gtp_{\lambda^1} \right) + \dotsb + \left(\setZ^d \cap \gtp_{\lambda^k} \right).\] By letting all $\lambda^i$ be equal, we recover the usual IDP.

This theorem does not generalize to skew shapes. The pair $\gtp_{3}$ and $\gtp_{2/1}$ yields a counterexample.

Gelfand–Tsetlin polytopes are special cases of marked order polytopes, which generalize order polytopes considered by R. Stanley [Sta86]. F. Ardila, T. Bliem, and D. Salazar generalize the bijection between order polytopes and chain polytopes and give a marked chain polytope analogue of GT-polytopes [ABS11]. In type $A$ these analogues are called Feigin–Fourier–Littelmann polytopes.

I. Makhlin studies marked chain-order polytopes for the Gelfand–Tsetlin poset [Mak24]. He realizes their toric varieties as explicit Gröbner/SAGBI degenerations of the flag variety, using pipe dreams to construct the Plücker-coordinate embeddings, and also obtains standard monomial theories and PBW-type monomial bases. X. Fang and G. Koshevoy compare Lusztig polytopes and FFLV polytopes [FK20].

O. Clarke, A. Higashitani, and F. Mohammadi relate Gelfand–Tsetlin and FFLV polytopes for Grassmannians to matching-field polytopes [CHM24]. They construct a sequence of combinatorial mutations between these polytopes that stays inside the class of matching-field polytopes, giving toric degenerations and Newton–Okounkov bodies for the Grassmannians.

R. G. Donnelly and M. W. Dunkum construct Gelfand–Tsetlin-type weight bases for all special linear Lie algebra representations whose characters are skew Schur functions [DD22]. Their bases give another representation-theoretic version of the GT-pattern picture for skew Schur objects. M. Fulmek enumerates symmetric Gelfand–Tsetlin patterns by linear algebra [Ful19]. H. Watanabe and K. Yamamura give an explicit type $A$ crystal isomorphism between the Lenart–Postnikov alcove-path model and the Gelfand–Tsetlin pattern model [WY21].

S. Daugherty, N. González, B. Muniz, P. S. Ocal, J. Pan, and J. Torres study the monoid algebra of semistandard tableaux [DGMO+25]. For $m=n$ this algebra agrees with the Gelfand–Tsetlin semigroup ring. They prove structural properties such as being Noetherian, reduced, Koszul, and Cohen–Macaulay, describe maximal ideals and irreducible modules, and relate the associated variety to toric degenerations of partial flag varieties.

The Gelfand–Tsetlin polytopes are also special cases of flow polytopes; see [LMD19]. In fact, R. I. Liu, Karola Mészáros, and A. S. Dizier develop a general connection between marked order polytopes and flow polytopes, with GT-polytopes as a central example. J. Jang and J. S. Kim compute volumes of flow polytopes related to caracol graphs [JK20].

#Birkhoff polytope as a GT-polytope

There is a natural correspondence between contingency matrices and certain GT-polytopes [Ale16]. A contingency matrix is a matrix of nonnegative integers with specified row sums and column sums.

Given a skew shape $\lambda/\mu$ and the GT-polytope for weight $\onevec = (1,1,\dotsc,1),$ each lattice point corresponds to a semistandard Young tableau. Let $a_{ij}$ denote the number of boxes with entry $j$ in row $i$ of the tableau. This gives an $m \times n$ matrix $A = (a_{ij})$ where each column sums to $1$ and row $i$ sums to $\lambda_i - \mu_i.$ The change of variables between GT-patterns and these matrices is an integral affine transformation, so lattice points in the GT-polytope are in bijection with contingency matrices with column sums $1$ and row sums $\lambda_i - \mu_i.$

The special case $\lambda_i - \mu_i = 1$ with $n = m$ recovers the Birkhoff polytope, which is the convex hull of all $n \times n$ permutation matrices. Since GT-polytopes are compressed (have the IDP), it follows that the Birkhoff polytope is also compressed, a fact proved independently in [Sta80]. There are several open questions about Birkhoff polytopes, such as how to compute their volumes; see [Pak00].

#Ehrhart polynomials

For a partition of length $m,$ \[\left| \setZ^d \cap \gtp_{\lambda} \right| = \schurS_{\lambda}(1^m) = \prod_{1 \leq i \lt j \leq m } \frac{\lambda_i - \lambda_j + j-i}{j-i},\] where the last identity is due to R. Stanley’s hook-content formula. Furthermore, the Ehrhart polynomial of $\gtp_{\lambda}$ is given by \[Ehr(\gtp_{\lambda};k) = \prod_{1 \leq i \lt j \leq m } \frac{k(\lambda_i - \lambda_j) + j-i}{j-i}.\]

Theorem (Jochemko–Menon [JM26, Thm. 3.5]).

The Ehrhart polynomial of $\gtp_{\lambda/\mu}$ has nonnegative coefficients. Equivalently, the function \[f(k) = \schurS_{k\lambda/k\mu}(1^m)\] is a polynomial in $k$ with nonnegative coefficients. This proves the conjecture of P. Alexandersson and E. Alhajjar in [AA18, Conj. 7].

The proof uses the ideal-chain decomposition for marked order polytopes, but the general closure criterion does not apply directly: ideals and filters of the GT poset lead to shifted skew shapes, whose Ehrhart positivity is not known. Jochemko and Menon instead show that the particular unmarked pieces in this decomposition are cell posets of ordinary skew shapes [JM26, Sec. 1 and proof of Thm. 3.5].

For example, $m=3,$ $\lambda=321$ and $\mu=1$ gives the Ehrhart polynomial \[Ehr(\gtp_{321/1},k) = \frac{1}{2} (k^5+6 k^4+14 k^3+16 k^2+9 k+2).\]

Their argument also gives positivity for the following unsliced flagged faces. In the paper’s indexing, let $\mathbf a,\mathbf b\in\setZ_{\geq0}^k$ be weakly increasing with $0\leq a_i\lt{}b_i\leq m+1,$ and restrict row $i$ to entries in $[a_i+1,b_i].$ The corresponding face is a marked order polytope obtained by identifying the coordinates forced equal, and it is Ehrhart positive [JM26, Sec. 3.2]. This does not settle the flagged Kostka positivity conjecture: fixing a weight intersects such a face with an affine slice, and the resulting rational polytope need not inherit coefficientwise Ehrhart positivity.

Adding an extra row to $\lambda/\mu$ gives \[\schurS_{k\lambda/k\mu}(1^n) = K_{k\nu/k\tau,(km)^n},\] for appropriate $\nu/\tau$ and $m = |\lambda/\mu|.$ The shape $\lambda/\mu$ can be filled freely, and the remaining boxes in the added row are then filled uniquely so that the weight becomes $m^n.$ A similar construction works for flagged skew Schur functions and flagged skew Kostka coefficients.

M. Bayer, B. Goeckner, S. J. Hong, T. McAllister, M. Olsen, C. Pinckney, J. Vega, and M. Yip study Newton polytopes arising from Schur polynomials and inflated symmetric Grothendieck polynomials [BGHM+21]. These polytopes have the integer decomposition property by saturated Newton polytope results, and the paper characterizes when they are reflexive.

Question

Do all Kogan faces of Gelfand–Tsetlin polytopes have nonnegative Ehrhart polynomial? P. Alexandersson gives an example of a (non-Kogan) face where the Ehrhart polynomial has a negative coefficient [Ale19]. However, this polynomial is of degree $20.$

#Vertices, $f$-vectors and differential equations

There has been some effort in counting the number of vertices of GT-polytopes. P. Gusev, V. A. Kiritchenko, and V. A. Timorin describe a generating function that enumerates the vertices of $\gtp_{\lambda}$ [GKT13].

Let $V(1^{i_1},2^{i_2},\dotsc,k^{i_k})$ denote the number of vertices of $\gtp_{\lambda}$ where $\lambda$ is the partition with type $(i_1,i_2,\dotsc,i_k).$ Consider the generating function \[E_k(z_1,\dotsc,z_k) \coloneqq \sum_{i_1,\dotsc,i_k \geq 0} V(1^{i_1},2^{i_2},\dotsc,k^{i_k}) \frac{z_1^{i_1}}{i_1!} \dotsm \frac{z_k^{i_k}}{i_k!}.\]

Theorem (See [GKT13]).

The formal power series $E_k(z_1,\dotsc,z_k)$ satisfies the partial differential equation \[\left( \frac{\partial^k}{\partial z_1 \dotsm \partial z_k} - \left(\frac{\partial}{\partial z_1}-\frac{\partial}{\partial z_2}\right) \dotsm \left(\frac{\partial}{\partial z_{k-1}}-\frac{\partial}{\partial z_k}\right) \right)E_k = 0\]

B. H. An, Y. Cho, and J. S. Kim later improve this result by giving a differential equation for the full $f$-vector of GT-polytopes [ACK18, Thm. 1.14]. They consider the generating function \[F_k(z_1,\dotsc,z_k;t) \coloneqq \sum_{d\geq 0} \sum_{i_1,\dotsc,i_k \geq 0} t^d f_d(1^{i_1},2^{i_2},\dotsc,k^{i_k}) \frac{z_1^{i_1}}{i_1!} \dotsm \frac{z_k^{i_k}}{i_k!}.\] where $f_d$ is the number of $d$-dimensional faces of $\gtp_{\lambda}.$

#Type $B$ and $C$

There are analogues of GT-polytopes for type $B$ and $C$; see [ABS11].

#Weight-restricted Gelfand–Tsetlin polytopes

Let $\gtp_{\lambda,w}\subseteq \gtp_{\lambda}$ be the weight-restricted Gelfand–Tsetlin polytope defined as $\gtp_{\lambda}$ but with the additional constraint that the difference of the row sums of rows $j+1$ and $j$ is exactly $w_j.$

By construction, lattice points in $\gtp_{\lambda,w}$ are in bijection with $\SSYT(\lambda,w).$ Therefore, the number of lattice points in $\gtp_{\lambda,w}$ is given by the Kostka coefficient $K_{\lambda w}.$

Analogous definitions apply to parallelogram GT-patterns and skew shapes.

Theorem (See [Kir01, p. 179]).

If $\lambda$ has $r$ parts and $\mu$ has $s$ parts, then the dimension of the polytope $\gtp_{\lambda,\mu}$ (assuming nonempty) is given by \[\dim \gtp_{\lambda,\mu} = (r-1)(s-1)-\binom{r}{2} - \sum_{i=1}^{r} \binom{\lambda'_i-\lambda'_{i+1}}{2}.\]

#Non-integrality and vertices

Contrary to $\gtp_{\lambda},$ the polytopes (also called Gelfand–Tsetlin polytopes) $\gtp_{\lambda,w}$ are not always integral. In fact, the denominators of the vertices can get arbitrarily large; see [DLM04].

P. Alexandersson proves that for $\lambda \vdash n,$ the polytope $\gtp_{\lambda,1^n}$ is nonintegral unless $\lambda=(2,2)$ or a hook shape [Ale16].

#Saturation theorem

The following theorem is a GT-polytope analogue of the celebrated Saturation theorem, proved by A. Knutson and T. Tao [KT99, KTW04]. They proved a similar statement for so-called Berenstein–Zelevinsky polytopes.

Theorem (See [Ale15]).

Every nonempty Gelfand–Tsetlin polytope $\gtp_{\lambda/\mu,w}$ contains at least one vertex with integer coordinates.

#Ehrhart polynomials and period collapse

Since the $\gtp_{\lambda/\mu,w}$ are not lattice polytopes, one does not expect the Ehrhart function to be a polynomial. However, these polytopes exhibit a period collapse [HM08] and they do have a polynomial Ehrhart function. See [Ras04] for a proof of this statement.

Conjecture (See [Kir01, p. 170] and later [KTT04]).

The coefficients of the Ehrhart polynomial of $\gtp_{\lambda/\mu,w}$ are nonnegative.

See also [Ale15] for related open conjectures.

One can also consider a fixed face of the Gelfand–Tsetlin polytope intersected with the hyperplane defined by a weight. Suppose $P$ is the polytope obtained from the non-skew shape $\lambda = (2,2,2)$ and $w = (1,1,\dotsc,1),$ intersected with the face requiring that $3$ is not in the first row of the corresponding SSYTs. Then the number of lattice points in $kP$ is \[3, 7, 13, 22, 34, 50, 70, 95, 125, 161, 203, \dotsc, \text{ for } k=1,2,\dotsc,\] which is a quasipolynomial of degree 3 with period 2.

#Interior lattice points and strict inequalities

The interior of a Gelfand–Tsetlin polytope consists of all points satisfying the interlacing inequalities strictly: \[x_{i+1,j} \gt{} x_{ij} \gt{} x_{i+1,j+1}.\] Integer points in the interior are strict GT-patterns, where every interlacing condition holds with strict inequality (cf. $SGT(\lambda)$ in Tokuyama’s formula).

By Ehrhart–Macdonald reciprocity, the number of interior lattice points in the $t$-th dilate satisfies \[\left| t \cdot \gtp^{\circ} \cap \setZ^d \right| = (-1)^d \cdot Ehr(\gtp; -t),\] where $d = \dim \gtp.$ This provides a method for computing the Ehrhart polynomial by evaluating lattice-point counts at both positive dilates (regular GT-patterns) and negative dilates (strict GT-patterns, via reciprocity).

For weight-restricted polytopes $\gtp_{\lambda/\mu,w},$ the interior lattice points satisfy strict interlacing together with the weight equalities (which are unchanged), and their count is again governed by Ehrhart–Macdonald reciprocity.

Example (Reciprocity).

Consider $\lambda = (5,3,0)$ with the triangular GT-pattern \[\begin{matrix} 5 && 3 && 0 \\ & a && b \\ && c \end{matrix} \qquad 5 \geq a \geq 3,\; 3 \geq b \geq 0,\; a \geq c \geq b.\] This polytope has dimension $3$ and contains $\schurS_{(5,3)}(1,1,1) = 42$ lattice points. For the interior, the strict conditions $5 \gt{} a \gt{} 3,$ $3 \gt{} b \gt{} 0,$ $a \gt{} c \gt{} b$ force $a = 4$ and $b \in \{1, 2\}$:

$b=1$: $c \in \{2, 3\}$;
$b=2$: $c = 3$;

giving $3$ interior lattice points. The Ehrhart polynomial is \[Ehr(\gtp_{(5,3,0)};t) = \frac{(2t+1)(5t+2)(3t+1)}{2},\] One verifies \[(-1)^3 \cdot Ehr(\gtp_{(5,3,0)};-1) = -\frac{(-1)(-3)(-2)}{2} = 3.\]

Example (Reciprocity in the Kostka case).

Consider instead the weight-restricted polytope $\gtp_{(3,2,1),(2,2,2)},$ whose lattice points are GT-patterns with top row $(3,2,1)$ and weight $(2,2,2)$: \[\begin{matrix} 3 && 2 && 1 \\ & a && b \\ && c \end{matrix} \qquad \begin{aligned} & 3 \geq a \geq 2,\; 2 \geq b \geq 1,\; a \geq c \geq b, \\ & a + b = 4,\; c = 2. \end{aligned}\] The weight conditions fix $c=2$ and $a+b=4.$ Combined with $3 \geq a \geq 2$ and $2 \geq b \geq 1,$ there are two solutions: $(a,b,c) \in \{(3,1,2),\; (2,2,2)\},$ confirming $K_{(3,2,1),(2,2,2)} = 2.$ The strict conditions $3 \gt{} a \gt{} 2,$ $2 \gt{} b \gt{} 1,$ $a \gt{} c \gt{} b$ have no integer solutions, so there are no interior lattice points. This reflects that the intersection with the weight hyperplanes is zero-dimensional.

#Littlewood–Richardson coefficients

The Littlewood–Richardson coefficient $c^\lambda_{\mu\nu}$ counts skew semistandard Young tableaux of shape $\lambda/\mu$ and content $\nu$ whose reverse reading word (read right to left across each row, from top to bottom) is a lattice word: at every position in the word, the number of occurrences of letter $i$ weakly exceeds the number of occurrences of letter $i+1,$ for all $i.$

In terms of GT-patterns, a skew tableau of shape $\lambda/\mu$ and content $\nu$ corresponds to a chain of partitions $\mu = \alpha^{(0)} \subseteq \alpha^{(1)} \subseteq \dotsb \subseteq \alpha^{(n)} = \lambda,$ where each $\alpha^{(k)}/\alpha^{(k-1)}$ is a horizontal strip of size $\nu_k.$ The quantity $\alpha^{(k)}_j - \alpha^{(k-1)}_j$ counts the cells with letter $k$ in row $j$ of the tableau.

#Yamanouchi inequalities on GT-patterns

The lattice word condition on the reverse reading word translates to the Yamanouchi inequalities, the following linear inequalities on the GT-pattern entries.

Proposition

A skew semistandard Young tableau with GT-pattern $\alpha^{(0)} \subseteq \alpha^{(1)} \subseteq \dotsb \subseteq \alpha^{(n)}$ is a Littlewood–Richardson tableau if and only if \[\sum_{l=1}^{j} \bigl(\alpha^{(k+1)}_l - \alpha^{(k)}_l\bigr) \;\leq\; \sum_{l=1}^{j-1} \bigl(\alpha^{(k)}_l - \alpha^{(k-1)}_l\bigr)\] for all $k = 1, \dotsc, n-1$ and $j = 1, \dotsc, m,$ where $m$ is the number of parts of $\lambda.$

Setting $j=1$ gives $\alpha^{(k+1)}_1 \leq \alpha^{(k)}_1$ for $k \geq 1,$ which combined with the interlacing condition $\alpha^{(k+1)}_1 \geq \alpha^{(k)}_1$ forces $\alpha^{(k+1)}_1 = \alpha^{(k)}_1$ for all $k \geq 1.$ This recovers the well-known fact that the first row of a Littlewood–Richardson tableau is filled entirely with letter $1.$

These additional linear constraints, imposed on $\gtp_{\lambda/\mu,\nu},$ cut out a polytope whose integer lattice points are in bijection with Littlewood–Richardson tableaux. In fact, this gives the description of Berenstein–Zelevinsky patterns.

Example (A Littlewood–Richardson case).

Consider $c^{(3,2,1)}_{(2,1),(2,1)}.$ The skew shape $(3,2,1)/(2,1)$ has three cells, at positions $(1,3),$ $(2,2),$ and $(3,1)$:

$ {}$ $ {}$ $ {}$

Since each row and column has at most one cell, every filling is semistandard. With content $(2,1)$ (two $1$s and one $2$), there are three fillings:

$ 1$ $ 1$ $ 2$ $ 1$ $ 2$ $ 1$ $ 2$ $ 1$ $ 1$

so $K_{(3,2,1)/(2,1), (2,1)} = 3.$ The corresponding GT-patterns have $\alpha^{(0)} = (2,1,0),$ $\alpha^{(2)} = (3,2,1),$ and the three possibilities for the middle row $\alpha^{(1)}$ are $(3,2,0),$ $(3,1,1),$ and $(2,2,1).$

Applying the Yamanouchi inequality with $k=1,$ $j=1$ gives $\alpha^{(2)}_1 - \alpha^{(1)}_1 \leq 0$:

$\alpha^{(1)} = (3,2,0)$: $3 - 3 = 0 \leq 0.$ Satisfied.
$\alpha^{(1)} = (3,1,1)$: $3 - 3 = 0 \leq 0.$ Satisfied.
$\alpha^{(1)} = (2,2,1)$: $3 - 2 = 1 \gt{} 0.$ Violated.

The third pattern is eliminated: it corresponds to the tableau with reverse reading word $2, 1, 1,$ which fails the lattice word condition at the first letter. The remaining inequalities (with $j=2,3$) are satisfied for the first two patterns, confirming $c^{(3,2,1)}_{(2,1),(2,1)} = 2.$ This agrees with the product decomposition \[\schurS_{(2,1)} \cdot \schurS_{(2,1)} = \schurS_{(4,2)} + \schurS_{(4,1,1)} + \schurS_{(3,3)} + 2\,\schurS_{(3,2,1)} + \schurS_{(3,1,1,1)} + \schurS_{(2,2,2)} + \schurS_{(2,2,1,1)}.\]

Example (A larger example).

For a larger example, consider $c^{(4,3,2)}_{(2,1),(3,2,1)},$ where the skew shape $(4,3,2)/(2,1)$ has six cells and content $\nu = (3,2,1).$ The GT-pattern has four rows: \[\alpha^{(0)} = (2,1,0) \subseteq \alpha^{(1)} \subseteq \alpha^{(2)} \subseteq \alpha^{(3)} = (4,3,2).\]

At step $0 \to 1$ (letter $1,$ strip size $3$), there are three extensions: \[\alpha^{(1)} \in \{(4,2,0),\; (4,1,1),\; (3,2,1)\}.\]
At step $1 \to 2$ (letter $2,$ strip size $2$), the Yamanouchi condition $\alpha^{(2)}_1 \leq \alpha^{(1)}_1$ eliminates extensions that place letter $2$ in the first row.
At step $2 \to 3$ (letter $3,$ strip size $1$), the Yamanouchi condition is checked against the cumulative row differences from the previous step.

After applying all constraints, one finds $c^{(4,3,2)}_{(2,1),(3,2,1)} = 2,$ compared to the skew Kostka number $K_{(4,3,2)/(2,1),(3,2,1)} = 6.$ The Yamanouchi conditions eliminate four of the six GT-patterns.

#Monotone triangles and alternating sign matrices

Monotone triangles are special triangular GT-patterns with additional constraints. First, the top row is $(n,n-1,\dotsc,1)$ and all rows must be strictly decreasing.

Monotone triangles of size $n$ are in bijection with $n \times n$ alternating sign matrices. Alternating sign matrices are matrices with entries in $\{-1,0,1\}$ such that in each row and column, the nonzero elements alternate in sign. Furthermore, the sum of the entries in each row (and column) is $1.$

Example

Consider the following $5\times 5$ alternating sign matrix, where we omit the zeros, and $-1$ is written as $\color{blue}{\overline{1}}.$ The rows are accumulated from the bottom, so that row $k$ from the bottom is the sum of the first $k$ rows. \[\begin{bmatrix} &1& & & \\ 1&\color{blue}{\overline{1}}& &1& \\ &1& &\color{blue}{\overline{1}}&1\\ & &1& & \\ & & &1& \end{bmatrix} \to \begin{bmatrix} 1&1&1&1&1 \\ 1& &1&1&1 \\ &1&1& &1 \\ & &1&1& \\ & & &1& \end{bmatrix}\] For each row, we then record the columns that contain a $1$ and write the column indices in a decreasing fashion. The result is a monotone triangle. \[\begin{matrix} 5&&4&&3&&2&&1 \\ &5&&4&&3&&1 \\ &&5&&3&&2 \\ &&&4&&3 \\ &&&&4& \\ \end{matrix}\]

D. Zeilberger [Zei95] proved that the number of monotone triangles of size $n,$ or equivalently the number of $n\times n$ alternating sign matrices, is given by \[\prod_{j=0}^{n-1}\frac{(3j+1)!}{(n+j)!}.\]

A generalization of this result is given in [Fis06], where difference operators are used in a formula to count the number of monotone triangles with prescribed top row $\lambda.$

#Applications

#Schur polynomials

The close connection between GT-patterns and Schur polynomials is explicit in the tableau formula. The Schur polynomial $\schurS_\lambda(x_1,\dotsc,x_n)$ can be expressed as a sum over $GT(\lambda),$ the set of triangular GT-patterns with top row equal to $\lambda.$ \[\schurS_\lambda(x_1,\dotsc,x_n) = \sum_{G \in GT(\lambda) } \xvec^{w(G)}\] where $w$ depends on the row-sums of the pattern $G.$

#Hall–Littlewood polynomials

The identity for Schur functions can be generalized to Hall–Littlewood polynomials, via a weighted version of Brion’s theorem.

Theorem (See [FM16]).

Let $\lambda$ be a partition. Then \[\hallLittlewoodP_\lambda(\xvec;t) = \sum_{G \in GT(\lambda) } p_G(t) \xvec^{w(G)}\] where $p_G(t)$ is a certain statistic that depends on $G.$

This theorem can also be proved by other methods; see [Mac95].

#Key polynomials

The key polynomials generalize the Schur polynomials. V. Kiritchenko, E. Smirnov, and V. Timorin prove a formula for key polynomials where the sum runs over lattice points in a union of certain faces of $GT(\lambda)$ [KST12]; see key polynomials from GT-patterns for details. The corresponding dilation series also has Lascoux’s tableauhedron interpretation in terms of left and right keys and compatible tableau chains. This is a weighted incidence construction, not another asserted convex-polytope model.

#Schubert polynomials

R. I. Liu, K. Mészáros, and A. S. Dizier prove that many Schubert polynomials can be obtained by projecting integer point transforms of Minkowski sums of GT-polytopes [LMD19]. More precisely, if $D$ is a column-convex diagram, then the character of the flagged Schur module of $D$ is obtained from the integer point transform of a convex polytope $P_D=GT(\lambda^{(1)})+\dotsb+GT(\lambda^{(n)})$ by a monomial specialization. When $D$ is the Rothe diagram of a column-convex permutation, this character is the corresponding Schubert polynomial.

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[Pak00]Igor Pak. Four questions on Birkhoff polytope. Annals of Combinatorics, 4(1):83–90, 2000.

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[Ras04]Etienne Rassart. A polynomiality property for Littlewood–Richardson coefficients. J. Comb. Theory Ser. A, 107(2):161–179, August 2004.

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[Sta86]Richard P. Stanley. A Baker’s dozen of conjectures concerning plane partitions. Combinatoire énumérative:285–293, 1986.

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[Sta86]Richard P. Stanley. Two poset polytopes. Discrete & Computational Geometry, 1(1):9–23, 1986.

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[Sta80]Richard P. Stanley. Decompositions of rational convex polytopes. Ann. Discrete Math.:333–342, 1980.

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[Sta01]Richard P. Stanley. Enumerative Combinatorics: Volume 2. Cambridge University Press, First, 2001.

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[WY21]Hideya Watanabe and Keita Yamamura. Alcove Paths and Gelfand–Tsetlin Patterns. Annals of Combinatorics, 25(3):645–676, 2021.

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[Zei95]Doron Zeilberger. Proof of the alternating sign matrix conjecture. The Electronic Journal of Combinatorics, 3(2), July 1995.

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The symmetric functions catalog